Abstract

We develop a theoretical model for the semiconductor generator of the sub-wavelength surface plasmons, operating on a single mode and often referred to as a spaser. We show that input-output characteristics of the single mode spaser does not exhibit nonlinearity inherent in most lasers, but the linewidth of the emission collapses, as in any laser which allows us to define the threshold. Our rigorous derivations show that as long as the mode remains substantially sub-wavelength in all three dimensions, the threshold current (power) shows virtually no dependence on the gain material and geometry of the active layer and is determined solely by the intrinsic loss of the metal in the device. For the semiconductor single mode surface plasmon generators operating in the telecommunication range the threshold current is on the order of 10-20 mA, and the threshold current density grows fast with the decrease of the device size reaching 100’s of kA/cm2 or more. This fact makes coherent sources of sub-wavelength SP’s unattainable from our point of view, but there exists a room for efficient broad-band incoherent SP sources either optically or electrically pumped.

© 2012 OSA

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2012

M. Khajavikhan, A. Simic, M. Katz, J. H. Lee, B. Slutsky, A. Mizrahi, V. Lomakin, and Y. Fainman, “Thresholdless nanoscale coaxial lasers,” Nature482(7384), 204–207 (2012).
[CrossRef] [PubMed]

R. F. Oulton, “Plasmonics: Loss and gain,” Nat. Photonics6(4), 219–221 (2012).
[CrossRef]

R. F. Oulton, “Surface plasmon lasers: sources of nanoscopic light,” Mater. Today15(1-2), 26–34 (2012).
[CrossRef]

R.-M. Ma, R.F. Oulton, V. J. Sorger, and X. Zhang, “Plasmon lasers: coherent light source at molecular scales,” Laser Photon. Rev.1–21 (2012).

J. B. Khurgin and G. Sun, “Practicality of compensating the loss in the plasmonic waveguides using semiconductor gain medium,” Appl. Phys. Lett.100(1), 011105 (2012).
[CrossRef]

K. Ding, Z. C. Liu, L. J. Yin, M. T. Hill, M. J. H. Marell, P. J. van Veldhoven, R. Nöetzel, and C. Z. Ning, “Room-temperature continuous wave lasing in deep-subwavelength metallic cavities under electrical injection,” Phys. Rev. B85(4), 041301 (2012).
[CrossRef]

2011

M. I. Stockman, “Spaser action, loss compensation, and stability in plasmonic systems with gain,” Phys. Rev. Lett.106(15), 156802 (2011).
[CrossRef] [PubMed]

G. Sun and J. B. Khurgin, “Plasmon Enhancement of Luminescence by Metal Nanoparticles,” IEEE J. Sel. Top. Quantum Electron.17(1), 110–118 (2011).
[CrossRef]

P. Berini and I. De Leon, “Surface plasmon–polariton amplifiers and lasers,” Nat. Photonics6(1), 16–24 (2011).
[CrossRef]

J. B. Khurgin and G. Sun, “Scaling of losses with size and wavelength in nanoplasmonics and metamaterials,” Appl. Phys. Lett.99(21), 211106 (2011).
[CrossRef]

C.-Y. Lu and S. L. Chuang, “A surface-emitting 3D metal-nanocavity laser: proposal and theory,” Opt. Express19(14), 13225–13244 (2011).
[CrossRef] [PubMed]

A. M. Lakhani, M.-K. Kim, E. K. Lau, and M. C. Wu, “Plasmonic crystal defect nanolaser,” Opt. Express19(19), 18237–18245 (2011).
[CrossRef] [PubMed]

J. H. Lee, M. Khajavikhan, A. Simic, Q. Gu, O. Bondarenko, B. Slutsky, M. P. Nezhad, and Y. Fainman, “Electrically pumped sub-wavelength metallo-dielectric pedestal pillar lasers,” Opt. Express19(22), 21524–21531 (2011).
[CrossRef] [PubMed]

2010

M. P. Nezhad, A. Simic, O. Bondarenko, B. Slutsky, A. Mizrahi, L. Feng, V. Lomakin, and Y. Fainman, “Room-temperature subwavelength metallo-dielectric lasers,” Nat. Photonics4(6), 395–399 (2010).
[CrossRef]

J. B. Khurgin and G. Sun, “In search of the elusive lossless metal,” Appl. Phys. Lett.96(18), 181102 (2010).
[CrossRef]

M. I. Stockman, “The spaser as a nanoscale quantum generator and ultrafast amplifier,” J. Opt.12(2), 024004 (2010).
[CrossRef]

S.-H. Kwon, J.-H. Kang, C. Seassal, S.-K. Kim, P. Regreny, Y.-H. Lee, C. M. Lieber, and H.-G. Park, “Subwavelength Plasmonic Lasing from a Semiconductor Nanodisk with Silver Nanopan Cavity,” Nano Lett.10(9), 3679–3683 (2010).
[CrossRef] [PubMed]

2009

M. A. Noginov, G. Zhu, A. M. Belgrave, R. Bakker, V. M. Shalaev, E. E. Narimanov, S. Stout, E. Herz, T. Suteewong, and U. Wiesner, “Demonstration of a spaser-based nanolaser,” Nature460(7259), 1110–1112 (2009).
[CrossRef] [PubMed]

R. F. Oulton, V. J. Sorger, T. Zentgraf, R.-M. Ma, C. Gladden, L. Dai, G. Bartal, and X. Zhang, “Plasmon lasers at deep subwavelength scale,” Nature461(7264), 629–632 (2009).
[CrossRef] [PubMed]

M. T. Hill, M. Marell, E. S. Leong, B. Smalbrugge, Y. Zhu, M. Sun, P. J. van Veldhoven, E. J. Geluk, F. Karouta, Y. S. Oei, R. Nötzel, C. Z. Ning, and M. K. Smit, “Lasing in metal-insulator-metal sub-wavelength plasmonic waveguides,” Opt. Express17(13), 11107–11112 (2009).
[CrossRef] [PubMed]

G. Sun, J. B. Khurgin, and R. A. Soref, “Practical enhancement of photoluminescence by metal nanoparticles,” Appl. Phys. Lett.94(10), 101103 (2009).
[CrossRef]

2008

M. I. Stockman, “Spasers explained,” Nat. Photonics2(6), 327–329 (2008).
[CrossRef]

M. Knight and N. J. Halas, “Nanoshells to nanoeggs to nanocups: optical properties of reduced symmetry core–shell nanoparticles beyond the quasistatic limit,” New J. Phys.10(10), 105006 (2008).
[CrossRef]

2006

F. Wang and Y. R. Shen, “General properties of local Plasmons in metal nanostructures,” Phys. Rev. Lett.97(20), 206806 (2006).
[CrossRef] [PubMed]

2005

A. V. Zayats, I. I. Smolyaninov, and A. A. Maradudin, “Nano-optics of surface plasmon polaritons,” Phys. Rep.408(3-4), 131–314 (2005).
[CrossRef]

2004

2003

D. J. Bergman and M. I. Stockman, “Surface Plasmon Amplification by Stimulated Emission of Radiation: Quantum Generation of Coherent Surface Plasmons in Nanosystems,” Phys. Rev. Lett.90(2), 027402 (2003).
[CrossRef] [PubMed]

2001

I. Vurgaftman, J. R. Meyer, and L.-R. Ram-Mohan, “Band parameters for III–V compound semiconductors and their alloys,” J. Appl. Phys.89(11), 5815–5875 (2001).
[CrossRef]

1999

O. Painter, R. K. Lee, A. Scherer, A. Yariv, J. D. O’Brien, P. D. Dapkus, and I. Kim, “Two-dimensional photonic band-gap defect mode laser,” Science284(5421), 1819–1821 (1999).
[CrossRef] [PubMed]

1995

G. H. C. New, “The origin of excess noise,” J. Mod. Opt.42(4), 799–810 (1995).
[CrossRef]

1994

G. Björk, A. Karlsson, and Y. Yamamoto, “Definition of a laser threshold,” Phys. Rev. A50(2), 1675–1680 (1994).
[CrossRef] [PubMed]

1992

S. L. McCall, A. F. J. Levi, R. E. Slusher, S. J. Pearton, and R. A. Logan, “Whispering‐gallery mode microdisk lasers,” Appl. Phys. Lett.60(3), 289–291 (1992).
[CrossRef]

1972

P. B. Johnson and R. Christy, “Optical constants of the noble metals,” Phys. Rev. B6(12), 4370–4379 (1972).
[CrossRef]

1970

I. Hayashi, M. B. Panish, P. W. Foy, and S. Sumski, “Junction lasers which operate continuously at room temperature,” Appl. Phys. Lett.17(3), 109–111 (1970).
[CrossRef]

1964

D. A. Kleiman, “The maser rate equations and spiking,” Bell Syst. Tech. J.43, 1505–1532 (1964).

1963

H. Kroemer, “A Proposed Class of Heterojunction Injection Lasers,” Proc. IEEE51(12), 1782–1783 (1963).
[CrossRef]

1962

R. N. Hall, G. E. Fenner, J. D. Kingsley, T. J. Soltys, and R. O. Carlson, “Coherent Light Emission From GaAs Junctions,” Phys. Rev. Lett.9(9), 366–368 (1962).
[CrossRef]

1958

A. L. Schawlow and C. H. Townes, “Infrared and optical masers,” Phys. Rev.112(6), 1940–1949 (1958).
[CrossRef]

1957

E. O. Kane, “Band structure of indium antimonide,” J. Phys. Chem. Solids1(4), 249–261 (1957).
[CrossRef]

Bakker, R.

M. A. Noginov, G. Zhu, A. M. Belgrave, R. Bakker, V. M. Shalaev, E. E. Narimanov, S. Stout, E. Herz, T. Suteewong, and U. Wiesner, “Demonstration of a spaser-based nanolaser,” Nature460(7259), 1110–1112 (2009).
[CrossRef] [PubMed]

Bartal, G.

R. F. Oulton, V. J. Sorger, T. Zentgraf, R.-M. Ma, C. Gladden, L. Dai, G. Bartal, and X. Zhang, “Plasmon lasers at deep subwavelength scale,” Nature461(7264), 629–632 (2009).
[CrossRef] [PubMed]

Belgrave, A. M.

M. A. Noginov, G. Zhu, A. M. Belgrave, R. Bakker, V. M. Shalaev, E. E. Narimanov, S. Stout, E. Herz, T. Suteewong, and U. Wiesner, “Demonstration of a spaser-based nanolaser,” Nature460(7259), 1110–1112 (2009).
[CrossRef] [PubMed]

Bergman, D. J.

D. J. Bergman and M. I. Stockman, “Surface Plasmon Amplification by Stimulated Emission of Radiation: Quantum Generation of Coherent Surface Plasmons in Nanosystems,” Phys. Rev. Lett.90(2), 027402 (2003).
[CrossRef] [PubMed]

Berini, P.

P. Berini and I. De Leon, “Surface plasmon–polariton amplifiers and lasers,” Nat. Photonics6(1), 16–24 (2011).
[CrossRef]

Björk, G.

G. Björk, A. Karlsson, and Y. Yamamoto, “Definition of a laser threshold,” Phys. Rev. A50(2), 1675–1680 (1994).
[CrossRef] [PubMed]

Bondarenko, O.

J. H. Lee, M. Khajavikhan, A. Simic, Q. Gu, O. Bondarenko, B. Slutsky, M. P. Nezhad, and Y. Fainman, “Electrically pumped sub-wavelength metallo-dielectric pedestal pillar lasers,” Opt. Express19(22), 21524–21531 (2011).
[CrossRef] [PubMed]

M. P. Nezhad, A. Simic, O. Bondarenko, B. Slutsky, A. Mizrahi, L. Feng, V. Lomakin, and Y. Fainman, “Room-temperature subwavelength metallo-dielectric lasers,” Nat. Photonics4(6), 395–399 (2010).
[CrossRef]

Carlson, R. O.

R. N. Hall, G. E. Fenner, J. D. Kingsley, T. J. Soltys, and R. O. Carlson, “Coherent Light Emission From GaAs Junctions,” Phys. Rev. Lett.9(9), 366–368 (1962).
[CrossRef]

Christy, R.

P. B. Johnson and R. Christy, “Optical constants of the noble metals,” Phys. Rev. B6(12), 4370–4379 (1972).
[CrossRef]

Chuang, S. L.

Dai, L.

R. F. Oulton, V. J. Sorger, T. Zentgraf, R.-M. Ma, C. Gladden, L. Dai, G. Bartal, and X. Zhang, “Plasmon lasers at deep subwavelength scale,” Nature461(7264), 629–632 (2009).
[CrossRef] [PubMed]

Dapkus, P. D.

O. Painter, R. K. Lee, A. Scherer, A. Yariv, J. D. O’Brien, P. D. Dapkus, and I. Kim, “Two-dimensional photonic band-gap defect mode laser,” Science284(5421), 1819–1821 (1999).
[CrossRef] [PubMed]

De Leon, I.

P. Berini and I. De Leon, “Surface plasmon–polariton amplifiers and lasers,” Nat. Photonics6(1), 16–24 (2011).
[CrossRef]

Ding, K.

K. Ding, Z. C. Liu, L. J. Yin, M. T. Hill, M. J. H. Marell, P. J. van Veldhoven, R. Nöetzel, and C. Z. Ning, “Room-temperature continuous wave lasing in deep-subwavelength metallic cavities under electrical injection,” Phys. Rev. B85(4), 041301 (2012).
[CrossRef]

Fainman, Y.

M. Khajavikhan, A. Simic, M. Katz, J. H. Lee, B. Slutsky, A. Mizrahi, V. Lomakin, and Y. Fainman, “Thresholdless nanoscale coaxial lasers,” Nature482(7384), 204–207 (2012).
[CrossRef] [PubMed]

J. H. Lee, M. Khajavikhan, A. Simic, Q. Gu, O. Bondarenko, B. Slutsky, M. P. Nezhad, and Y. Fainman, “Electrically pumped sub-wavelength metallo-dielectric pedestal pillar lasers,” Opt. Express19(22), 21524–21531 (2011).
[CrossRef] [PubMed]

M. P. Nezhad, A. Simic, O. Bondarenko, B. Slutsky, A. Mizrahi, L. Feng, V. Lomakin, and Y. Fainman, “Room-temperature subwavelength metallo-dielectric lasers,” Nat. Photonics4(6), 395–399 (2010).
[CrossRef]

M. P. Nezhad, K. Tetz, and Y. Fainman, “Gain assisted propagation of surface plasmon polaritons on planar metallic waveguides,” Opt. Express12(17), 4072–4079 (2004).
[CrossRef] [PubMed]

Feng, L.

M. P. Nezhad, A. Simic, O. Bondarenko, B. Slutsky, A. Mizrahi, L. Feng, V. Lomakin, and Y. Fainman, “Room-temperature subwavelength metallo-dielectric lasers,” Nat. Photonics4(6), 395–399 (2010).
[CrossRef]

Fenner, G. E.

R. N. Hall, G. E. Fenner, J. D. Kingsley, T. J. Soltys, and R. O. Carlson, “Coherent Light Emission From GaAs Junctions,” Phys. Rev. Lett.9(9), 366–368 (1962).
[CrossRef]

Foy, P. W.

I. Hayashi, M. B. Panish, P. W. Foy, and S. Sumski, “Junction lasers which operate continuously at room temperature,” Appl. Phys. Lett.17(3), 109–111 (1970).
[CrossRef]

Geluk, E. J.

Gladden, C.

R. F. Oulton, V. J. Sorger, T. Zentgraf, R.-M. Ma, C. Gladden, L. Dai, G. Bartal, and X. Zhang, “Plasmon lasers at deep subwavelength scale,” Nature461(7264), 629–632 (2009).
[CrossRef] [PubMed]

Gu, Q.

Halas, N. J.

M. Knight and N. J. Halas, “Nanoshells to nanoeggs to nanocups: optical properties of reduced symmetry core–shell nanoparticles beyond the quasistatic limit,” New J. Phys.10(10), 105006 (2008).
[CrossRef]

Hall, R. N.

R. N. Hall, G. E. Fenner, J. D. Kingsley, T. J. Soltys, and R. O. Carlson, “Coherent Light Emission From GaAs Junctions,” Phys. Rev. Lett.9(9), 366–368 (1962).
[CrossRef]

Hayashi, I.

I. Hayashi, M. B. Panish, P. W. Foy, and S. Sumski, “Junction lasers which operate continuously at room temperature,” Appl. Phys. Lett.17(3), 109–111 (1970).
[CrossRef]

Herz, E.

M. A. Noginov, G. Zhu, A. M. Belgrave, R. Bakker, V. M. Shalaev, E. E. Narimanov, S. Stout, E. Herz, T. Suteewong, and U. Wiesner, “Demonstration of a spaser-based nanolaser,” Nature460(7259), 1110–1112 (2009).
[CrossRef] [PubMed]

Hill, M. T.

K. Ding, Z. C. Liu, L. J. Yin, M. T. Hill, M. J. H. Marell, P. J. van Veldhoven, R. Nöetzel, and C. Z. Ning, “Room-temperature continuous wave lasing in deep-subwavelength metallic cavities under electrical injection,” Phys. Rev. B85(4), 041301 (2012).
[CrossRef]

M. T. Hill, M. Marell, E. S. Leong, B. Smalbrugge, Y. Zhu, M. Sun, P. J. van Veldhoven, E. J. Geluk, F. Karouta, Y. S. Oei, R. Nötzel, C. Z. Ning, and M. K. Smit, “Lasing in metal-insulator-metal sub-wavelength plasmonic waveguides,” Opt. Express17(13), 11107–11112 (2009).
[CrossRef] [PubMed]

Johnson, P. B.

P. B. Johnson and R. Christy, “Optical constants of the noble metals,” Phys. Rev. B6(12), 4370–4379 (1972).
[CrossRef]

Kane, E. O.

E. O. Kane, “Band structure of indium antimonide,” J. Phys. Chem. Solids1(4), 249–261 (1957).
[CrossRef]

Kang, J.-H.

S.-H. Kwon, J.-H. Kang, C. Seassal, S.-K. Kim, P. Regreny, Y.-H. Lee, C. M. Lieber, and H.-G. Park, “Subwavelength Plasmonic Lasing from a Semiconductor Nanodisk with Silver Nanopan Cavity,” Nano Lett.10(9), 3679–3683 (2010).
[CrossRef] [PubMed]

Karlsson, A.

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S. L. McCall, A. F. J. Levi, R. E. Slusher, S. J. Pearton, and R. A. Logan, “Whispering‐gallery mode microdisk lasers,” Appl. Phys. Lett.60(3), 289–291 (1992).
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S.-H. Kwon, J.-H. Kang, C. Seassal, S.-K. Kim, P. Regreny, Y.-H. Lee, C. M. Lieber, and H.-G. Park, “Subwavelength Plasmonic Lasing from a Semiconductor Nanodisk with Silver Nanopan Cavity,” Nano Lett.10(9), 3679–3683 (2010).
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M. Khajavikhan, A. Simic, M. Katz, J. H. Lee, B. Slutsky, A. Mizrahi, V. Lomakin, and Y. Fainman, “Thresholdless nanoscale coaxial lasers,” Nature482(7384), 204–207 (2012).
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S. L. McCall, A. F. J. Levi, R. E. Slusher, S. J. Pearton, and R. A. Logan, “Whispering‐gallery mode microdisk lasers,” Appl. Phys. Lett.60(3), 289–291 (1992).
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M. Khajavikhan, A. Simic, M. Katz, J. H. Lee, B. Slutsky, A. Mizrahi, V. Lomakin, and Y. Fainman, “Thresholdless nanoscale coaxial lasers,” Nature482(7384), 204–207 (2012).
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R.-M. Ma, R.F. Oulton, V. J. Sorger, and X. Zhang, “Plasmon lasers: coherent light source at molecular scales,” Laser Photon. Rev.1–21 (2012).

R. F. Oulton, V. J. Sorger, T. Zentgraf, R.-M. Ma, C. Gladden, L. Dai, G. Bartal, and X. Zhang, “Plasmon lasers at deep subwavelength scale,” Nature461(7264), 629–632 (2009).
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J. B. Khurgin and G. Sun, “Practicality of compensating the loss in the plasmonic waveguides using semiconductor gain medium,” Appl. Phys. Lett.100(1), 011105 (2012).
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G. Sun and J. B. Khurgin, “Plasmon Enhancement of Luminescence by Metal Nanoparticles,” IEEE J. Sel. Top. Quantum Electron.17(1), 110–118 (2011).
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J. B. Khurgin and G. Sun, “Scaling of losses with size and wavelength in nanoplasmonics and metamaterials,” Appl. Phys. Lett.99(21), 211106 (2011).
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Suteewong, T.

M. A. Noginov, G. Zhu, A. M. Belgrave, R. Bakker, V. M. Shalaev, E. E. Narimanov, S. Stout, E. Herz, T. Suteewong, and U. Wiesner, “Demonstration of a spaser-based nanolaser,” Nature460(7259), 1110–1112 (2009).
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K. Ding, Z. C. Liu, L. J. Yin, M. T. Hill, M. J. H. Marell, P. J. van Veldhoven, R. Nöetzel, and C. Z. Ning, “Room-temperature continuous wave lasing in deep-subwavelength metallic cavities under electrical injection,” Phys. Rev. B85(4), 041301 (2012).
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F. Wang and Y. R. Shen, “General properties of local Plasmons in metal nanostructures,” Phys. Rev. Lett.97(20), 206806 (2006).
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M. A. Noginov, G. Zhu, A. M. Belgrave, R. Bakker, V. M. Shalaev, E. E. Narimanov, S. Stout, E. Herz, T. Suteewong, and U. Wiesner, “Demonstration of a spaser-based nanolaser,” Nature460(7259), 1110–1112 (2009).
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K. Ding, Z. C. Liu, L. J. Yin, M. T. Hill, M. J. H. Marell, P. J. van Veldhoven, R. Nöetzel, and C. Z. Ning, “Room-temperature continuous wave lasing in deep-subwavelength metallic cavities under electrical injection,” Phys. Rev. B85(4), 041301 (2012).
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A. V. Zayats, I. I. Smolyaninov, and A. A. Maradudin, “Nano-optics of surface plasmon polaritons,” Phys. Rep.408(3-4), 131–314 (2005).
[CrossRef]

Zentgraf, T.

R. F. Oulton, V. J. Sorger, T. Zentgraf, R.-M. Ma, C. Gladden, L. Dai, G. Bartal, and X. Zhang, “Plasmon lasers at deep subwavelength scale,” Nature461(7264), 629–632 (2009).
[CrossRef] [PubMed]

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R.-M. Ma, R.F. Oulton, V. J. Sorger, and X. Zhang, “Plasmon lasers: coherent light source at molecular scales,” Laser Photon. Rev.1–21 (2012).

R. F. Oulton, V. J. Sorger, T. Zentgraf, R.-M. Ma, C. Gladden, L. Dai, G. Bartal, and X. Zhang, “Plasmon lasers at deep subwavelength scale,” Nature461(7264), 629–632 (2009).
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M. A. Noginov, G. Zhu, A. M. Belgrave, R. Bakker, V. M. Shalaev, E. E. Narimanov, S. Stout, E. Herz, T. Suteewong, and U. Wiesner, “Demonstration of a spaser-based nanolaser,” Nature460(7259), 1110–1112 (2009).
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Appl. Phys. Lett.

J. B. Khurgin and G. Sun, “Scaling of losses with size and wavelength in nanoplasmonics and metamaterials,” Appl. Phys. Lett.99(21), 211106 (2011).
[CrossRef]

J. B. Khurgin and G. Sun, “Practicality of compensating the loss in the plasmonic waveguides using semiconductor gain medium,” Appl. Phys. Lett.100(1), 011105 (2012).
[CrossRef]

I. Hayashi, M. B. Panish, P. W. Foy, and S. Sumski, “Junction lasers which operate continuously at room temperature,” Appl. Phys. Lett.17(3), 109–111 (1970).
[CrossRef]

J. B. Khurgin and G. Sun, “In search of the elusive lossless metal,” Appl. Phys. Lett.96(18), 181102 (2010).
[CrossRef]

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